μ-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

Journal of Singularities
volume 18 (2018), 119-214

Received: 12 October 2017. Accepted: 27 February 2018.

DOI: 10.5427/jsing.2018.18i

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Abstract:

This paper is a sequel to our previous work, in which a notion of marking of isolated hypersurface singularities was defined, and a moduli space M_μ^{mar} for marked singularities in one μ-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a μ-constant monodromy group G^{mar}\subset G_\Z. Here G_\Z is the group of automorphisms of a Milnor lattice which respect the Seifert form.

It was conjectured that M_μ^{mar} is connected. This is equivalent to G^{mar}= G_\Z. Also Torelli type conjectures were formulated. In our earlier work, M_\mu^{mar}, G_\Z and G^{mar} were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture that G^{mar}= G_\Z and M_μ^{mar} is connected does not hold for certain subseries of the bimodal series.

Keywords:

μ-constant monodromy group, marked singularity, moduli space, Torelli type problem, quadrangle singularities, bimodal series

Mathematical Subject Classification:

[2000] 32S15, 32S40, 14D22, 58K70

Author(s) information: